Have done it with United, going through the contact form on their website. Got a very helpful reply after about a week, containing all the information I asked for: Tail number, wheels off and wheels on time.

All the things you probably hate about travelling are warm reminders that I'm home

I think the best way to get a registration is to look at ACARS logs available onlime. I think you'll find that a good number of European and trans-Atlantic flights are in here (other areas have people logging too).